Introduction to Economic Analysis | Flat World Knowledge

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15.3 Effect of Taxes

Learning Objective

How does a monopoly respond to taxes?

A tax imposed on a seller with monopoly power performs differently than a tax imposed on a competitive industry. Ultimately, a perfectly competitive industry must pass on all of a tax to consumers because, in the long run, the competitive industry earns zero profits. In contrast, a monopolist might absorb some portion of a tax even in the long run.

To model the effect of taxes on a monopoly, consider a monopolist who faces a tax rate t per unit of sales. This monopolist earns $π = p (q) q - c (q) - t q .$

The first-order condition for profit maximization yields $0 = \frac{\partial π}{\partial q} = p (q_{m}) + q_{m} p^{'} (q_{m}) - c^{'} (q_{m}) - t .$

Viewing the monopoly quantity as a function of t, we obtain $\frac{d q_{m}}{d t} = \frac{1}{2 p^{'} (q_{m}) + q_{m} p^{″} (q_{m}) - c^{″} (q_{m})} < 0$ with the sign following from the second-order condition for profit maximization. In addition, the change in price satisfies $p^{'} (q_{m}) \frac{d q_{m}}{d t} = \frac{p^{'} (q_{m})}{2 p^{'} (q_{m}) + q_{m} p^{″} (q_{m}) - c^{″} (q_{m})} > 0 .$

Thus, a tax causes a monopoly to increase its price. In addition, the monopoly price rises by less than the tax if $p^{'} (q_{m}) \frac{d q_{m}}{d t} < 1,$ or $p^{'} (q_{m}) + q_{m} p^{″} (q_{m}) - c^{″} (q_{m}) < 0 .$

This condition need not be true but is a standard regularity condition imposed by assumption. It is true for linear demand and increasing marginal cost. It is false for constant elasticity of demand, ε > 1 (which is the relevant case, for otherwise the second-order conditions fail), and constant marginal cost. In the latter case (constant elasticity and marginal cost), a tax on a monopoly increases price by more than the amount of the tax.

Key Takeaways

A perfectly competitive industry must pass on all of a tax to consumers because, in the long run, the competitive industry earns zero profits. A monopolist might absorb some portion of a tax even in the long run.
A tax causes a monopoly to increase its price and reduce its quantity.
A tax may or may not increase the monopoly markup.

Exercises

Use a revealed preference argument to show that a per-unit tax imposed on a monopoly causes the quantity to fall. That is, hypothesize quantities qb before the tax and qa after the tax, and show that two facts—the before-tax monopoly preferred qb to qa, and the taxed monopoly made higher profits from qb—together imply that qb ≤ qa.
When both demand and supply have constant elasticity, use the results of 0 to compute the effect of a proportional tax (i.e., a portion of the price paid to the government).

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